The q-deformed conformal matrix model partition function, after a Fourier transform, is a Toda tau-function whose shifted ratios satisfy the discrete Painleve q-PVI equation, with the string equation supplied by Virasoro constraints.
A $q$-anaolg of the sixth Painlev\'e equation
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abstract
A $q$-difference analog of the sixth Painlev\'e equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear differential equations. The continuous limit and special solutions in terms of $q$-hypergeometric functions are also discussed.
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Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models
The q-deformed conformal matrix model partition function, after a Fourier transform, is a Toda tau-function whose shifted ratios satisfy the discrete Painleve q-PVI equation, with the string equation supplied by Virasoro constraints.